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(a) For a positive, decreasing function $f:[1,?)?R$ that is integrable on every $[a,b]?[1,?)$, define $a_{n}=k=1?n?f(k)??_{1}f(x)dx.$ Show that ${a_{n}}$ is nonincreasing, bounded, and $0<n??lim?a_{n}<f(1)$. (Hint: note that $?_{1}f(x)dx=k=2?n??_{k?1}f(x)dx=k=1?n?1??_{k}f(x)dx.)$ (b) Recall that the harmonic numbers ${H_{n}}$ are defined $H_{n}=k=1?n?k1?,n?1.$ Show that Euler's constant $?=n??lim?(H_{n}?gn)$ exists, and that $0<?<1$. (Hint: use part (a) with $f(x)=1/x$.)

5.a) To show that {a_n} is non-increasing, we can show that for all n, an?an+1. We have:an=?k=1nf(k)??1nf(x)dxan+1=?k=1n+1f(k)??1n+1f(x)dxNow, by subt